Difficulty: Intermediate to Advanced | Prerequisites: Utility Functions and Consumer Choice notes (Doc 1). You must be comfortable finding optimum bundles for Cobb-Douglas, perfect substitutes and perfect complements before starting here.
This is where consumer theory gets computational. You are taking the demand functions from the previous topic and transforming them into tools for measuring welfare: how much better or worse off is a consumer when prices change? The indirect utility function, expenditure function, compensating variation and equivalent variation are all tightly linked, and exam questions typically ask you to work through them in sequence.
The indirect utility function (IUF) tells you the maximum utility at given prices and income. The expenditure function (EF) tells you the minimum cost to reach a given utility level. Compensating variation and equivalent variation use these two functions to put a dollar figure on how a price change affects welfare. The substitution and income effects decompose the total change in demand into a "pure price" component and a "purchasing power" component.
Indirect Utility Function (IUF)
The utility function with the demand functions substituted in, so it expresses maximum utility as a function of prices and income: V(Px, Py, M). Think of it as: given these prices and this budget, what is the best I can do?
Expenditure Function (EF)
The inverse of the IUF. It tells you the minimum income needed to reach a given utility level V at given prices: E(Px, Py, V). Think of it as: how much would I need to spend to be this well off?
Compensating Variation (CV)
The amount of extra money a consumer would need after a price increase to be as well off as before. Uses the original utility level and the new prices. In simple terms, how much cash would compensate you for the price hike?
Equivalent Variation (EV)
The amount of money you could take away from a consumer at the original prices that would leave them as badly off as the price increase does. Uses the new utility level and the original prices. In simple terms, how much income loss at old prices is equivalent to the damage from the price change?
Substitution effect
The change in quantity demanded due purely to the change in relative prices, holding utility constant. Measured as Xc - X*, where Xc is the compensated demand and X* is the original demand.
Income effect
The change in quantity demanded due to the change in purchasing power (real income), after accounting for the substitution effect. Measured as X** - Xc, where X** is the new demand.
Compensated budget constraint
A hypothetical budget line at the new prices but with enough income to achieve the original utility level. It is parallel to the new budget line and tangent to the original indifference curve.
The IUF is found by substituting the demand functions (x*, y*) back into the utility function.
Cobb-Douglas: U = x^(1/2) y^(1/2)
Demand: x = M/(2Px), y = M/(2Py)
V = (M/2Px)^(1/2) × (M/2Py)^(1/2) = M / sqrt(4PxPy)
If the utility function has a coefficient (e.g. U = 3x^(1/2)y^(1/2)), that coefficient multiplies the entire IUF.
Perfect Complements: U = Min(ax, by)
Set ax = by, substitute into the budget constraint to get x = M/(Px + (b/a)Py).
V = aM / (Px + (b/a)Py)
For Min(2x, y): V = 2M / (Px + 2Py)
For Min(3x, y): V = 3M / (Px + 3Py)
Perfect Substitutes: U = ax + by
First find the switching price by setting MRS = MRT: a/b = Px/Py.
If Px/Py < a/b (MRS > MRT), the consumer buys all x: V = aM/Px
If Px/Py > a/b (MRT > MRS), the consumer buys all y: V = bM/Py
If Px/Py = a/b, any bundle on the budget line works
For U = 6x + y with Py = 1: the switching price is Px = 6. When Px < 6, V = 6M/Px. When Px > 6, V = M/Py.
Note: if the utility function is U = 2x + 3y, do not forget to multiply by the coefficient when computing utility. If the consumer buys all x, utility is 2(M/Px) = 2M/Px, not just M/Px.
The expenditure function is the algebraic inverse of the IUF: solve V = f(Px, Py, M) for M.
Cobb-Douglas: V = M / sqrt(4PxPy)
E = V × sqrt(4PxPy)
Perfect Complements: V = aM / (Px + (b/a)Py)
E = V(Px + (b/a)Py) / a
For Min(2x, y): E = V(Px + 2Py) / 2
Perfect Substitutes
Depends on which good is purchased:
If buying all x: V = aM/Px, so E = VPx/a
If buying all y: V = bM/Py, so E = VPy/b
The EF is the key bridge between utility levels and dollar amounts, which is what makes CV and EV calculations possible.
CV answers: how much extra money does the consumer need at the new prices to reach the original utility level?
Step-by-step method:
Calculate original utility using OG prices and OG income in the IUF
Plug that utility level and the new prices into the expenditure function to find the required income
CV = required income - original income
The mnemonic: OG M, OG prices into IUF to get OG utility, then sub new prices into EF.
Worked example: Perfect Complements
U = Min(2x, y), M = 12, Py = 1, Px goes from 1 to 2.
IUF: V = 2M/(Px + 2Py) = 2(12)/(1 + 2) = 8
EF at new prices: E = V(Px + 2Py)/2 = 8(2 + 2)/2 = 16
CV = 16 - 12 = 4
The consumer needs an extra $4 to maintain utility of 8 at the new prices.
Worked example: Cobb-Douglas
U = x^(1/2)y^(1/2), M = 3600, Py = 1, Px goes from 1 to 9.
IUF: V = 3600/sqrt(4 × 1 × 1) = 3600/2 = 1800. Wait: let us redo with the formula V = M/sqrt(4PxPy). V = 3600/sqrt(4) = 1800. But the source notes give V = 900 here. Checking: V = 3600/sqrt(4 × 1 × 1) = 3600/2 = 1800 at original prices. Actually the source computes 3600/sqrt(16) for some steps. Let us follow the source's own numbers carefully.
At OG prices (Px=1, Py=1): V = 3600/sqrt(4×1×1) = 3600/2 = 1800. Wait, the source says V = 900. Looking again: the source says V = 3600/sqrt(16) = 900, which uses Px = 4. The source problem #3 may have Px initially = 1 but then the separate Q1 exam problem has Px = 4. Let us use the cleaner problem #3.
Problem 3: M = 36 (not 3600). Px initially 1, rises to 9. Using the IUF V = M/sqrt(4PxPy):
V = 36/sqrt(4×1×1) = 36/2 = 18
EF at new prices: E = V × sqrt(4×9×1) = 18 × sqrt(36) = 18 × 6 = 108
CV = 108 - 36 = 72
EV answers: how much money could you take away from the consumer at the original prices to make them exactly as badly off as the price change does?
Step-by-step method:
Calculate new utility using OG income and new prices in the IUF
Plug that new utility level and the original prices into the expenditure function
EV = original income - required income at old prices
The mnemonic: OG M and new prices into IUF to get new utility, then sub old prices into EF.
Worked example: Perfect Complements
U = Min(2x, y), M = 12, Py = 1, Px goes from 1 to 2.
New utility: V = 2(12)/(2 + 2) = 6
EF at OG prices: E = 6(1 + 2)/2 = 9
EV = 12 - 9 = 3
Key difference between CV and EV
CV uses the original utility level and asks what income is needed at new prices. EV uses the new utility level and asks what income would be needed at old prices. For a price increase, CV > EV for normal goods.
Measure | Utility level used | Prices in the EF |
|---|---|---|
CV | Original (before price change) | New prices |
EV | New (after price change) | Old prices |
When a price changes, the total change in quantity demanded can be split into two parts.
Substitution effect (Xc - X)*
The change in demand due purely to the change in relative prices, holding utility constant. You find Xc by using the compensated budget constraint: the new prices with enough income to achieve the original utility.
Calculate the compensated income using the EF: plug original utility and new prices into the expenditure function
Find the compensated demand Xc using the demand function at new prices and compensated income
Substitution effect = Xc - X* (original demand)
The compensated budget constraint is parallel to the new budget line (same price ratio) and tangent to the original indifference curve (same utility).
Income effect (X - Xc)**
The remaining change in demand, due to the shift in purchasing power.
X** is the final demand at the new prices and original income
Income effect = X** - Xc
Worked example: Perfect Complements
U = Min(2x, y), M = 12, Px goes from 1 to 2, Py = 1.
Original demand: 12 = x + 2x = 3x, so X* = 4.
Compensated income (from CV section): E = 16 at new prices for original utility of 8. At new prices: 16 = 2x + 2x = 4x, so Xc = 4.
Substitution effect = 4 - 4 = 0. (Perfect complements always have zero substitution effect because the consumer stays at the kink.)
New demand at original income: 12 = 2x + 2x = 4x, so X** = 3.
Income effect = 3 - 4 = -1.
Worked example: Cobb-Douglas
U = x^(1/2)y^(1/2), M = 3600, Px goes from 4 to 9, Py = 1.
Original demand: X* = 3600/(2×4) = 450.
Compensated income at new prices: first find original V = 3600/sqrt(4×4×1) = 3600/4 = 900. Then E = 900 × sqrt(4×9×1) = 900 × 6 = 5400.
Xc = 5400/(2×9) = 300. Substitution effect = 300 - 450 = -150.
X** at new prices and original income: X** = 3600/(2×9) = 200. Income effect = 200 - 300 = -100.
Total change = -150 + (-100) = -250, which matches X** - X* = 200 - 450 = -250.
Students often mix up which prices go where in CV vs EV. Remember: CV uses new prices in the EF (because you are compensating at the new price regime). EV uses old prices in the EF (because you are finding the equivalent income loss at the old prices).
The substitution effect for perfect complements is always zero. Students waste time trying to compute it. Because the consumer is always at the kink, changing relative prices without changing utility does not change the bundle.
For perfect substitutes, the substitution effect can be enormous (the consumer may switch entirely from one good to the other) or zero (if they stay on the same good). Students often forget that a price change can trigger a regime switch.
Students sometimes confuse the sign convention. For a price increase of good x, both the substitution and income effects should reduce demand for x (for a normal good). If your numbers show demand increasing after a price rise, check your work.
⚠️ The IUF, EF, CV, EV, substitution effect and income effect sequence is a near-certainty on the exam. Practise it end to end for each utility type.
⚠️ For perfect substitutes, check whether the price change crosses the switching price. This determines whether the consumer changes goods entirely, which changes every subsequent calculation.
⚠️ The income (M) never changes in these problems. What changes is prices. M stays constant throughout; the compensated income is a hypothetical construct for decomposing effects.
⚠️ Double-check the direction of CV and EV. For a price increase, CV is positive (consumer needs more money) and EV is positive (consumer would accept losing that much money as equivalent).
True or false: Compensating variation uses the new utility level. (False. CV uses the original utility level.)
Fill in the blank: The substitution effect is measured as ________ minus ________. (Xc minus X*)
True or false: For perfect complements, the substitution effect is always zero. (True.)
Fill in the blank: The expenditure function is the ________ of the indirect utility function. (Inverse)
True or false: If Px increases and the good is normal, the income effect should increase demand for x. (False. For a normal good, the income effect of a price increase reduces demand.)
Q: U = Min(2x, y), M = 12, Py = 1, Px rises from 1 to 2. Find the IUF, EF, CV, EV, substitution effect and income effect.
A: IUF: V = 2M/(Px + 2Py). EF: E = V(Px + 2Py)/2. CV: V = 2(12)/(1+2) = 8; E = 8(2+2)/2 = 16; CV = 16-12 = 4. EV: V = 2(12)/(2+2) = 6; E = 6(1+2)/2 = 9; EV = 12-9 = 3. Substitution effect: Xc - X* = 4 - 4 = 0. Income effect: X** - Xc = 3 - 4 = -1.
Q: U = x^(1/2)y^(1/2), M = 3600, Py = 1, Px = 4. Find the optimum bundle and utility.
A: MRS = MRT gives y/x = 4/1, so y = 4x. Budget: 3600 = 4x + 4x = 8x, x = 450, y = 1800. V = 3600/sqrt(4 × 4 × 1) = 3600/4 = 900.
Q: For the problem above, Px rises to 9. Find the compensating variation.
A: Original V = 900. EF at new prices: E = 900 × sqrt(4 × 9 × 1) = 900 × 6 = 5400. CV = 5400 - 3600 = 1800.
Q: Same problem. Find the equivalent variation.
A: New V = 3600/sqrt(4 × 9 × 1) = 3600/6 = 600. EF at old prices: E = 600 × sqrt(4 × 4 × 1) = 600 × 4 = 2400. EV = 3600 - 2400 = 1200.
Q: Same problem. Find the substitution and income effects.
A: X* = 450. Compensated income at new prices = 5400. Xc = 5400/(2 × 9) = 300. Substitution effect = 300 - 450 = -150. X** = 3600/(2 × 9) = 200. Income effect = 200 - 300 = -100. Total = -250.
Q: U = 6x + y, M = 60, Py = 1, Px rises from 1 to 10. Find CV.
A: Switching price: MRS = 6 = Px/Py, so Px = 6. At Px = 1 (< 6), buy all x: V = 6(60)/1 = 360. At Px = 10 (> 6), buy all y: IUF = M/Py. For CV, need original utility 360 at new prices. At new prices, consumer buys all y, so V = M/Py, meaning M = 360 × 1 = 360. CV = 360 - 60 = 300.
This material builds directly on the utility functions and consumer choice covered in Doc 1. If your demand functions or MRS calculations are shaky, revisit those notes first.
CV and EV connect forward to welfare economics and policy analysis: they are the tools economists use to measure whether a tax, subsidy or price change makes consumers better or worse off, in dollar terms.
The substitution and income effect decomposition reappears in labour economics (the backward-bending labour supply curve) and in the analysis of Giffen goods, where the income effect is large enough to overwhelm the substitution effect.
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